Question

A mouthwash is \(21.6 \%\) ethyl alcohol by mass. If each bottle contains \(0.358\) pt of mouthwash with a density of \(0.876 \mathrm{~g} / \mathrm{mL}\), how many kilograms of ethyl alcohol are in 180 bottles of the mouthwash?

Short answer

5.75208 kg

Step-by-step solution

- Convert pints to milliliters

Since each bottle contains 0.358 pints of mouthwash, use the conversion factor: 1 pint = 473.176 milliliters. Compute the volume in milliliters as follows:\(0.358 \text{ pt} \times 473.176 \text{ mL/pt} = 169.093 \text{ mL}\)

- Calculate the mass of mouthwash in one bottle

Use the density of the mouthwash (0.876 g/mL) and the volume in milliliters from Step 1 (169.093 mL) to find the mass in grams:\(169.093 \text{ mL} \times 0.876 \text{ g/mL} = 148.020 \text{ g}\)

- Find the mass of ethyl alcohol in one bottle

The mass of ethyl alcohol in one bottle can be found using the percentage by mass of ethyl alcohol (21.6\text{ %}):\(148.020 \text{ g} \times 0.216 = 31.956 \text{ g}\)

- Calculate the total mass of ethyl alcohol in 180 bottles

Multiply the mass of ethyl alcohol in one bottle by the total number of bottles (180):\(31.956 \text{ g} \times 180 = 5752.08 \text{ g}\)

- Convert grams to kilograms

Since 1 kilogram = 1000 grams, convert the total mass of ethyl alcohol to kilograms:\(5752.08 \text{ g} / 1000 = 5.75208 \text{ kg}\)

Key concepts

Density

Density is a measure of how much mass is contained in a given volume. It is typically expressed in units such as grams per cubic centimeter (g/cm\textsuperscript{3}) or grams per milliliter (g/mL). In this exercise, the density of the mouthwash helps us determine its mass from its volume.

To find mass using density, we use the formula:

\[ \text{Mass} = \text{Density} \times \text{Volume} \]

In the solution, we know the density of the mouthwash (\text{0.876 g/mL}) and the volume in milliliters. By multiplying these two values, we can find the mass of the mouthwash in one bottle.

Mass Percentage

Mass percentage tells us how much of one substance is present in a mixture, based on the mass of the mixture. It is expressed as a percentage.

In the exercise, the mouthwash contains \text{21.6%} ethyl alcohol by mass. This means that for every \text{100 grams} of mouthwash, \text{21.6 grams} are ethyl alcohol.

To find the mass of ethyl alcohol in a bottle of mouthwash, we use the mass percentage formula:

\[ \text{Mass of Ethyl Alcohol} = \text{Mass of Mouthwash} \times \frac{\text{Mass Percentage of Ethyl Alcohol}}{100} \]

Using this formula, we can calculate the mass of ethyl alcohol in one bottle by multiplying the total mass of mouthwash by \text{0.216}, which is the decimal form of \text{21.6%}.

Unit Conversion

Unit conversion is essential in chemistry to ensure consistency and accuracy of measurements. When working with different units, we often need to convert them to match the desired unit of the final answer.

In the exercise, we start with a volume in pints (pt) and need to convert it to milliliters (mL). The conversion factor is:

\[ \text{1 pint} = 473.176 \text{ milliliters} \]

We use this factor to convert the volume of mouthwash from pints to milliliters:

\[ 0.358 \text{ pt} \times 473.176 \text{ mL/pt} = 169.093 \text{ mL} \]

Another conversion we perform is from grams (g) to kilograms (kg), since the final mass of ethyl alcohol is needed in kilograms. The conversion factor is:

\[ \text{1 kilogram} = 1000 \text{ grams} \]

By dividing the total mass in grams by \text{1000}, we convert it to kilograms.

Mass Calculation

Mass calculation involves combining data and formulas to determine the mass of a substance in a specific context. In this problem, we start by calculating the mass of mouthwash per bottle, followed by the mass of ethyl alcohol in one bottle, and finally scaling it up for \text{180 bottles}.

First, we find the mass of the mouthwash in one bottle using density and volume:

\[ 169.093 \text{ mL} \times 0.876 \text{ g/mL} = 148.020 \text{ g} \]

Next, we use the mass percentage to find the mass of ethyl alcohol in one bottle:

\[ 148.020 \text{ g} \times 0.216 = 31.956 \text{ g} \]

Finally, we calculate the total mass of ethyl alcohol in \text{180 bottles}:

\[ 31.956 \text{ g} \times 180 = 5752.08 \text{ g} \]

And convert this to kilograms:

\[ 5752.08 \text{ g} / 1000 = 5.75208 \text{ kg} \]